The form of the partial fraction decomposition of a rational function is given below. x − 3 x 2 − 26 ( x + 1 ) ( x 2 + 9 ) = A x + 1 + B x + C x 2 + 9 What are the values of A , B and C?So can someone explain how to get the answer. I'm not sure how I got to the answer but I know B = 0 and C = 1I also found the indefinite integral, which equals 1 3 tan − 1 ⁡ ( x 3 ) − 3 log ⁡ ( 1 + x ) + C

Question

The form of the partial fraction decomposition of a rational function is given below.            x      −      3              x        2            −      26              (      x      +      1      )      (              x        2            +      9      )        =      A          x      +      1        +            B      x      +      C                      x        2            +      9      What are the values of   A  ,  B and   C?So can someone explain how to get the answer. I&#039;m not sure how I got to the answer but I know   B  =  0 and   C  =  1I also found the indefinite integral, which equals      1    3        tan          −      1        ⁡      (          x      3        )    −  3  log  ⁡  (  1  +  x  )  +  C

Lamont Adkins · Accepted Answer

Multiplying either sides by    (  x  +  1  )  (      x    2    +  9  )  ,  x  −  3      x    2    −  26  =  A  (      x    2    +  9  )  +  (  B  x  +  C  )  (  x  +  1  )    ⟹    −  3      x    2    +  x  −  26  =      x    2    (  A  +  B  )  +  x  (  B  +  C  )  +  9  A  +  CCompare the constants and the coefficients of   x  ,      x    2   of the above identity to form three simultaneous equations with three unknowns A,B,C

George Bray · Accepted Answer

Multiply by   x  +  1 and let   x  →  −  1 to obtain            −      1      −      3      −      26              (      −      1              )                  2                    +      9        =  A    ⟹    A  =  −  3.Multiply both sides by   x, let   x  →  ∞, and use   A  =  −  3 to obtain  −  3  =  A  +  B    ⟹    B  =  0.Finally, evaluate at   x  =  0, and use   A  =  −  27      /    9:            −      26        9    =  A  +      C    9      ⟹    C  =  9      (    −          26      9        +          27      9        )    =  1.